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G = C2xD4.D7order 224 = 25·7

Direct product of C2 and D4.D7

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2xD4.D7, D4.7D14, C14:2SD16, C28.16D4, C28.13C23, Dic14:6C22, C7:C8:8C22, C7:3(C2xSD16), (C2xD4).4D7, (D4xC14).3C2, (C2xC4).48D14, C14.46(C2xD4), (C2xC14).40D4, C4.6(C7:D4), (C2xDic14):9C2, (C7xD4).7C22, C4.13(C22xD7), (C2xC28).31C22, C22.22(C7:D4), (C2xC7:C8):5C2, C2.10(C2xC7:D4), SmallGroup(224,128)

Series: Derived Chief Lower central Upper central

C1C28 — C2xD4.D7
C1C7C14C28Dic14C2xDic14 — C2xD4.D7
C7C14C28 — C2xD4.D7
C1C22C2xC4C2xD4

Generators and relations for C2xD4.D7
 G = < a,b,c,d,e | a2=b4=c2=d7=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >

Subgroups: 222 in 68 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2xC4, C2xC4, D4, D4, Q8, C23, C14, C14, C14, C2xC8, SD16, C2xD4, C2xQ8, Dic7, C28, C2xC14, C2xC14, C2xSD16, C7:C8, Dic14, Dic14, C2xDic7, C2xC28, C7xD4, C7xD4, C22xC14, C2xC7:C8, D4.D7, C2xDic14, D4xC14, C2xD4.D7
Quotients: C1, C2, C22, D4, C23, D7, SD16, C2xD4, D14, C2xSD16, C7:D4, C22xD7, D4.D7, C2xC7:D4, C2xD4.D7

Smallest permutation representation of C2xD4.D7
On 112 points
Generators in S112
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 106)(72 107)(73 108)(74 109)(75 110)(76 111)(77 112)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)
(1 22 8 15)(2 23 9 16)(3 24 10 17)(4 25 11 18)(5 26 12 19)(6 27 13 20)(7 28 14 21)(29 43 36 50)(30 44 37 51)(31 45 38 52)(32 46 39 53)(33 47 40 54)(34 48 41 55)(35 49 42 56)(57 71 64 78)(58 72 65 79)(59 73 66 80)(60 74 67 81)(61 75 68 82)(62 76 69 83)(63 77 70 84)(85 106 92 99)(86 107 93 100)(87 108 94 101)(88 109 95 102)(89 110 96 103)(90 111 97 104)(91 112 98 105)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 43)(37 44)(38 45)(39 46)(40 47)(41 48)(42 49)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)(22 23 24 25 26 27 28)(29 30 31 32 33 34 35)(36 37 38 39 40 41 42)(43 44 45 46 47 48 49)(50 51 52 53 54 55 56)(57 58 59 60 61 62 63)(64 65 66 67 68 69 70)(71 72 73 74 75 76 77)(78 79 80 81 82 83 84)(85 86 87 88 89 90 91)(92 93 94 95 96 97 98)(99 100 101 102 103 104 105)(106 107 108 109 110 111 112)
(1 96 8 89)(2 95 9 88)(3 94 10 87)(4 93 11 86)(5 92 12 85)(6 98 13 91)(7 97 14 90)(15 103 22 110)(16 102 23 109)(17 101 24 108)(18 100 25 107)(19 99 26 106)(20 105 27 112)(21 104 28 111)(29 68 36 61)(30 67 37 60)(31 66 38 59)(32 65 39 58)(33 64 40 57)(34 70 41 63)(35 69 42 62)(43 75 50 82)(44 74 51 81)(45 73 52 80)(46 72 53 79)(47 71 54 78)(48 77 55 84)(49 76 56 83)

G:=sub<Sym(112)| (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,106)(72,107)(73,108)(74,109)(75,110)(76,111)(77,112)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105), (1,22,8,15)(2,23,9,16)(3,24,10,17)(4,25,11,18)(5,26,12,19)(6,27,13,20)(7,28,14,21)(29,43,36,50)(30,44,37,51)(31,45,38,52)(32,46,39,53)(33,47,40,54)(34,48,41,55)(35,49,42,56)(57,71,64,78)(58,72,65,79)(59,73,66,80)(60,74,67,81)(61,75,68,82)(62,76,69,83)(63,77,70,84)(85,106,92,99)(86,107,93,100)(87,108,94,101)(88,109,95,102)(89,110,96,103)(90,111,97,104)(91,112,98,105), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,43)(37,44)(38,45)(39,46)(40,47)(41,48)(42,49)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98), (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28)(29,30,31,32,33,34,35)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49)(50,51,52,53,54,55,56)(57,58,59,60,61,62,63)(64,65,66,67,68,69,70)(71,72,73,74,75,76,77)(78,79,80,81,82,83,84)(85,86,87,88,89,90,91)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112), (1,96,8,89)(2,95,9,88)(3,94,10,87)(4,93,11,86)(5,92,12,85)(6,98,13,91)(7,97,14,90)(15,103,22,110)(16,102,23,109)(17,101,24,108)(18,100,25,107)(19,99,26,106)(20,105,27,112)(21,104,28,111)(29,68,36,61)(30,67,37,60)(31,66,38,59)(32,65,39,58)(33,64,40,57)(34,70,41,63)(35,69,42,62)(43,75,50,82)(44,74,51,81)(45,73,52,80)(46,72,53,79)(47,71,54,78)(48,77,55,84)(49,76,56,83)>;

G:=Group( (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,106)(72,107)(73,108)(74,109)(75,110)(76,111)(77,112)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105), (1,22,8,15)(2,23,9,16)(3,24,10,17)(4,25,11,18)(5,26,12,19)(6,27,13,20)(7,28,14,21)(29,43,36,50)(30,44,37,51)(31,45,38,52)(32,46,39,53)(33,47,40,54)(34,48,41,55)(35,49,42,56)(57,71,64,78)(58,72,65,79)(59,73,66,80)(60,74,67,81)(61,75,68,82)(62,76,69,83)(63,77,70,84)(85,106,92,99)(86,107,93,100)(87,108,94,101)(88,109,95,102)(89,110,96,103)(90,111,97,104)(91,112,98,105), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,43)(37,44)(38,45)(39,46)(40,47)(41,48)(42,49)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98), (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28)(29,30,31,32,33,34,35)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49)(50,51,52,53,54,55,56)(57,58,59,60,61,62,63)(64,65,66,67,68,69,70)(71,72,73,74,75,76,77)(78,79,80,81,82,83,84)(85,86,87,88,89,90,91)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112), (1,96,8,89)(2,95,9,88)(3,94,10,87)(4,93,11,86)(5,92,12,85)(6,98,13,91)(7,97,14,90)(15,103,22,110)(16,102,23,109)(17,101,24,108)(18,100,25,107)(19,99,26,106)(20,105,27,112)(21,104,28,111)(29,68,36,61)(30,67,37,60)(31,66,38,59)(32,65,39,58)(33,64,40,57)(34,70,41,63)(35,69,42,62)(43,75,50,82)(44,74,51,81)(45,73,52,80)(46,72,53,79)(47,71,54,78)(48,77,55,84)(49,76,56,83) );

G=PermutationGroup([[(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,50),(16,51),(17,52),(18,53),(19,54),(20,55),(21,56),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49),(57,85),(58,86),(59,87),(60,88),(61,89),(62,90),(63,91),(64,92),(65,93),(66,94),(67,95),(68,96),(69,97),(70,98),(71,106),(72,107),(73,108),(74,109),(75,110),(76,111),(77,112),(78,99),(79,100),(80,101),(81,102),(82,103),(83,104),(84,105)], [(1,22,8,15),(2,23,9,16),(3,24,10,17),(4,25,11,18),(5,26,12,19),(6,27,13,20),(7,28,14,21),(29,43,36,50),(30,44,37,51),(31,45,38,52),(32,46,39,53),(33,47,40,54),(34,48,41,55),(35,49,42,56),(57,71,64,78),(58,72,65,79),(59,73,66,80),(60,74,67,81),(61,75,68,82),(62,76,69,83),(63,77,70,84),(85,106,92,99),(86,107,93,100),(87,108,94,101),(88,109,95,102),(89,110,96,103),(90,111,97,104),(91,112,98,105)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,50),(30,51),(31,52),(32,53),(33,54),(34,55),(35,56),(36,43),(37,44),(38,45),(39,46),(40,47),(41,48),(42,49),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97),(91,98)], [(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21),(22,23,24,25,26,27,28),(29,30,31,32,33,34,35),(36,37,38,39,40,41,42),(43,44,45,46,47,48,49),(50,51,52,53,54,55,56),(57,58,59,60,61,62,63),(64,65,66,67,68,69,70),(71,72,73,74,75,76,77),(78,79,80,81,82,83,84),(85,86,87,88,89,90,91),(92,93,94,95,96,97,98),(99,100,101,102,103,104,105),(106,107,108,109,110,111,112)], [(1,96,8,89),(2,95,9,88),(3,94,10,87),(4,93,11,86),(5,92,12,85),(6,98,13,91),(7,97,14,90),(15,103,22,110),(16,102,23,109),(17,101,24,108),(18,100,25,107),(19,99,26,106),(20,105,27,112),(21,104,28,111),(29,68,36,61),(30,67,37,60),(31,66,38,59),(32,65,39,58),(33,64,40,57),(34,70,41,63),(35,69,42,62),(43,75,50,82),(44,74,51,81),(45,73,52,80),(46,72,53,79),(47,71,54,78),(48,77,55,84),(49,76,56,83)]])

C2xD4.D7 is a maximal subgroup of
D28.2D4  D4.D7:C4  Dic7:6SD16  Dic14:2D4  Dic14.D4  D4.6D28  D14:SD16  C7:C8:1D4  D4.D28  D4.1D28  C42.51D14  D4.2D28  D28:17D4  Dic14:17D4  C7:C8:23D4  C7:C8:5D4  C42.61D14  C42.214D14  C42.65D14  C42.74D14  Dic14:9D4  C28:4SD16  (C2xD8).D7  C56:11D4  C56.22D4  Dic14:D4  Dic7:3SD16  C56.31D4  Dic14:7D4  C56:15D4  M4(2).13D14  (C7xD4).31D4  (C7xD4).32D4  C2xD7xSD16  D8:6D14  D28.33C23
C2xD4.D7 is a maximal quotient of
C4:C4.231D14  C28.38SD16  D4.2D28  C4:D4.D7  Dic14:17D4  C7:C8:23D4  C28.16D8  Dic14:9D4  C28:4SD16  C28.SD16  C28.11Q16  Dic14:6Q8  (C7xD4).31D4

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D7A7B7C8A8B8C8D14A···14I14J···14U28A···28F
order1222224444777888814···1414···1428···28
size111144222828222141414142···24···44···4

44 irreducible representations

dim11111222222224
type++++++++++-
imageC1C2C2C2C2D4D4D7SD16D14D14C7:D4C7:D4D4.D7
kernelC2xD4.D7C2xC7:C8D4.D7C2xDic14D4xC14C28C2xC14C2xD4C14C2xC4D4C4C22C2
# reps11411113436666

Matrix representation of C2xD4.D7 in GL4(F113) generated by

112000
011200
0010
0001
,
112000
011200
0001
001120
,
112000
10100
0001
0010
,
106000
21600
0010
0001
,
668100
694700
0010013
001313
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[112,0,0,0,0,112,0,0,0,0,0,112,0,0,1,0],[112,10,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[106,2,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[66,69,0,0,81,47,0,0,0,0,100,13,0,0,13,13] >;

C2xD4.D7 in GAP, Magma, Sage, TeX

C_2\times D_4.D_7
% in TeX

G:=Group("C2xD4.D7");
// GroupNames label

G:=SmallGroup(224,128);
// by ID

G=gap.SmallGroup(224,128);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,96,218,579,159,69,6917]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^7=1,e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations

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